Ricci flow with surgery on three-manifolds.

*(English)*Zbl 1130.53002This is the second paper written by G. Perelman which proves the geometrization conjecture. It is presented by the author as a “technical paper” in which some assertions made in [G. Perelman, “The entropy formula for the Ricci flow and its geometric applications”, arXiv: math.DG/0211159 (2002; Zbl 1130.53001)] are proved and some others are corrected. It presents the Ricci flow with surgery on three-manifolds which is a version of the Ricci flow taking into account the singularities. It is inspired by the construction made by R. Hamilton in [Commun. Anal. Geom. 5, No. 1, 1–92 (1997; Zbl 0892.53018)]. As for G. Perelman (loc. cit.) the details can be found in [B. Kleiner and J. Lott, “Notes on Perelman’s papers”, arXiv: math.DG/0605667 (2006), J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3 (2007; Zbl 1179.57045) and H.-D. Cao and X.-P. Zhu, Asian J. Math. 10, No. 2, 165–492 (2006; Zbl 1200.53057)].

Sections 1 and 2: These sections present preliminary material on ancient and standard solutions. In section 1 the author gives a more precise classification of ancient solutions by the study of their asymptotic soliton. Let us recall that the ancient solutions are those which have an infinite past and ought to be (with some more properties) the infinitesimal models for the singularities. The asymptotic soliton is a limit when time goes to \(-\infty\) of a rescaling of the ancient solution. These notions are defined in section 11 of Perelman (loc. cit.) and more refined properties are described here.

Section 2 is devoted to the standard solution. This is a solution of the Ricci flow whose initial data is a half cylinder capped-off by a hemisphere. The initial data being non compact existence and uniqueness of the solution is not immediate. It is shown that the Ricci flow with such an initial data exists in the time interval \([0, 1)\) and is unique. An extremely detailed proof is given in [Morgan and Tian (loc. cit.), see also Kleiner and Lott (loc. cit.) and Cao and Zhu (loc. cit.)].

Section 3: In this important section the first time for which the flow becomes singular is described. Being singular means that the scalar curvature becomes infinite somewhere. If it is infinite everywhere then the manifold is covered by canonical neighbourhoods and its topology is thus completely known. If it becomes infinite on a subset \(\Omega\), it is open and one can describe the part of \(\Omega\) of high scalar curvature. They are, again, covered with canonical neighbourhoods. This leads to the important notion of horns which are subsets of \(\Omega\) diffeomorphic to open half cylinders and on which the scalar curvature goes to infinity on one end. It is in these horns that the surgery will take place. The subset \(M\setminus\Omega\) is covered by canonical neighbourhoods and is topologically simple (and known).

Sections 4 and 5: These are the sections in which the Ricci flow with surgery is defined. In section 4 the so-called Ricci flow with cutoff appears. It is given by a collection of smooth Ricci flows defined on a 3-manifold on adjacent intervals of time. On the boundary of these intervals surgeries take place. In this section it is supposed that the flow satisfies the a priori assumptions which are: the Hamilton-Ivey pinching property and the canonical neighbourhood property for points whose scalar curvature is larger than a parameter \(r > 0\). The surgery is done in horns. More precisely it is shown that if one goes “far enough” into a horn then one finds a neck of size \(2/\delta\) for some parameter \(\delta> 0\) small enough and of curvature close to \(h > 0\) another parameter depending on \(r\) and \(\delta\). The flow is called a Ricci flow with \(\delta\)-cutoff. Now, this \(\delta\)-neck is cut in the middle and a suitably rescaled compact piece of the standard solution is glued on one side. The flow may start again with the new Riemannian manifold thus obtained. An important property is the fact that the added piece (called an almost-standard cap) remains close to the evoluting standard solution for a while.

In section 5 the author shows that the flow can be defined, satisfying the a priori assumptions for all time. The parameters \(r\) and \(\delta\) must now depend on the time parameter and one issue is to prove that they do not go to zero in finite time. The proof is very close to the proof of the canonical neighbourhood theorem done in [Perelman (loc. cit.)] section 12, taking into account the surgeries. This is the key result of this series of work by G. Perelman.

Sections 6 and 7: Now, the long time behaviour of the Ricci flow with surgery is analysed carefully. Section 6 presents technical issues such as curvature estimates in the future and the past of a given time slice.

Section 7 defines and describes the thick-thin decomposition of the manifold. It is inspired by R. Hamilton [Commun. Anal. Geom. 7, No. 4, 695–729 (1999; Zbl 0939.53024)]. It is shown that thick parts become more and more hyperbolic bounded by incompressible tori. The description of the thin part relies on an unpublished paper by the author; the conclusion is that it is a graph manifold.

Section 8: It contains an alternative approach. The geometric decomposition is described using the values of a Riemannian invariant as threshold for the different possibilities. This invariant is the first eigenvalue of a Schrödinger operator whose potential is given by the scalar curvature. A simpler argument can be found in [Kleiner and Lott (loc. cit.), section 93].

At the time when this review is written the fact that the thin part is a graph manifold is still a bit controversial although it is widely believed to be true. Some details are missing in the literature. An alternative approach is given in [L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti, “Suites de métriques extraites du flot de Ricci sur les variétés asphériques de dimension 3”, arXiv:0706.2065].

Sections 1 and 2: These sections present preliminary material on ancient and standard solutions. In section 1 the author gives a more precise classification of ancient solutions by the study of their asymptotic soliton. Let us recall that the ancient solutions are those which have an infinite past and ought to be (with some more properties) the infinitesimal models for the singularities. The asymptotic soliton is a limit when time goes to \(-\infty\) of a rescaling of the ancient solution. These notions are defined in section 11 of Perelman (loc. cit.) and more refined properties are described here.

Section 2 is devoted to the standard solution. This is a solution of the Ricci flow whose initial data is a half cylinder capped-off by a hemisphere. The initial data being non compact existence and uniqueness of the solution is not immediate. It is shown that the Ricci flow with such an initial data exists in the time interval \([0, 1)\) and is unique. An extremely detailed proof is given in [Morgan and Tian (loc. cit.), see also Kleiner and Lott (loc. cit.) and Cao and Zhu (loc. cit.)].

Section 3: In this important section the first time for which the flow becomes singular is described. Being singular means that the scalar curvature becomes infinite somewhere. If it is infinite everywhere then the manifold is covered by canonical neighbourhoods and its topology is thus completely known. If it becomes infinite on a subset \(\Omega\), it is open and one can describe the part of \(\Omega\) of high scalar curvature. They are, again, covered with canonical neighbourhoods. This leads to the important notion of horns which are subsets of \(\Omega\) diffeomorphic to open half cylinders and on which the scalar curvature goes to infinity on one end. It is in these horns that the surgery will take place. The subset \(M\setminus\Omega\) is covered by canonical neighbourhoods and is topologically simple (and known).

Sections 4 and 5: These are the sections in which the Ricci flow with surgery is defined. In section 4 the so-called Ricci flow with cutoff appears. It is given by a collection of smooth Ricci flows defined on a 3-manifold on adjacent intervals of time. On the boundary of these intervals surgeries take place. In this section it is supposed that the flow satisfies the a priori assumptions which are: the Hamilton-Ivey pinching property and the canonical neighbourhood property for points whose scalar curvature is larger than a parameter \(r > 0\). The surgery is done in horns. More precisely it is shown that if one goes “far enough” into a horn then one finds a neck of size \(2/\delta\) for some parameter \(\delta> 0\) small enough and of curvature close to \(h > 0\) another parameter depending on \(r\) and \(\delta\). The flow is called a Ricci flow with \(\delta\)-cutoff. Now, this \(\delta\)-neck is cut in the middle and a suitably rescaled compact piece of the standard solution is glued on one side. The flow may start again with the new Riemannian manifold thus obtained. An important property is the fact that the added piece (called an almost-standard cap) remains close to the evoluting standard solution for a while.

In section 5 the author shows that the flow can be defined, satisfying the a priori assumptions for all time. The parameters \(r\) and \(\delta\) must now depend on the time parameter and one issue is to prove that they do not go to zero in finite time. The proof is very close to the proof of the canonical neighbourhood theorem done in [Perelman (loc. cit.)] section 12, taking into account the surgeries. This is the key result of this series of work by G. Perelman.

Sections 6 and 7: Now, the long time behaviour of the Ricci flow with surgery is analysed carefully. Section 6 presents technical issues such as curvature estimates in the future and the past of a given time slice.

Section 7 defines and describes the thick-thin decomposition of the manifold. It is inspired by R. Hamilton [Commun. Anal. Geom. 7, No. 4, 695–729 (1999; Zbl 0939.53024)]. It is shown that thick parts become more and more hyperbolic bounded by incompressible tori. The description of the thin part relies on an unpublished paper by the author; the conclusion is that it is a graph manifold.

Section 8: It contains an alternative approach. The geometric decomposition is described using the values of a Riemannian invariant as threshold for the different possibilities. This invariant is the first eigenvalue of a Schrödinger operator whose potential is given by the scalar curvature. A simpler argument can be found in [Kleiner and Lott (loc. cit.), section 93].

At the time when this review is written the fact that the thin part is a graph manifold is still a bit controversial although it is widely believed to be true. Some details are missing in the literature. An alternative approach is given in [L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti, “Suites de métriques extraites du flot de Ricci sur les variétés asphériques de dimension 3”, arXiv:0706.2065].

Reviewer: Gérard Besson (Grenoble)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

57M40 | Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |

57R60 | Homotopy spheres, Poincaré conjecture |